Paper 2012/498

Almost Perfect Algebraic Immune Functions with Good Nonlinearity

Meicheng Liu and Dongdai Lin

Abstract

In the last decade, algebraic and fast algebraic attacks are regarded as the most successful attacks on LFSR-based stream ciphers. Since the notion of algebraic immunity was introduced, the properties and constructions of Boolean functions with maximum algebraic immunity have been researched in a large number of papers. However, there are few results with respect to Boolean functions with provable good immunity against fast algebraic attacks. In previous literature, only Carlet-Feng function, which is affine equivalent to discrete logarithm function, was proven to be optimal against fast algebraic attacks as well as algebraic attacks. In this paper, it is proven that a family of $2k$-variable Boolean functions, including the function recently constructed by Tang et al. [IEEE TIT 59(1): 653--664, 2013], are almost perfect algebraic immune for any integer $k\geq 3$. More exactly, they achieve optimal algebraic immunity and almost perfect immunity to fast algebraic attacks. The functions of such family are balanced and have optimal algebraic degree. A lower bound on their nonlinearity is obtained based on the work of Tang et al., which is better than that of Carlet-Feng function. It is also checked for $3\leq k\leq 9$ that the exact nonlinearity of such functions is very good, which is slightly smaller than that of Carlet-Feng function, and some functions of this family even have a slightly larger nonlinearity than Tang et al.'s function. To sum up, among the known functions with provable good immunity against fast algebraic attacks, the functions of this family make a trade-off between the exact value and the lower bound of nonlinearity.